This tag is for questions about the foundations of mathematics, and the formalization of mathematical concepts in foundational theories (e.g. set theory, category theory, and type theory).

learn more… | top users | synonyms

1
vote
3answers
113 views

Book on foundational reasoning of standard arithmetics “curriculum”

I am interested in a book that is about arithmetics but the presentation is not just the known to all formulas but the foundational logic behind it. The closest example I can think about is the way ...
-1
votes
2answers
41 views

What is the denial of a statement in logic math?

I'm trying to get the hang of denials in logic in math. I would like to use these two examples: "Some people are honest and some people are not honest. (All people)" "No one loves everybody. (All ...
0
votes
1answer
45 views

Can one use the Hilbert-Ackermann Consistency Theorem to prove the consistency of $PRA$?

In his textbook Mathematical Logic, Shoenfield states the Hilbert-Ackermann Consistency Theorem as follows: "Consistency Theorem (Hilbert-Ackermann): An open theory $T$ is inconsistent iff there is ...
3
votes
1answer
84 views

Theorems not Formulable in Set Theory

Several sites I have been reading say that set theory is a good foundation for mathematics because virtually every theorem can be cast into a theorem in set theory. What is an example of a theorem ...
-2
votes
4answers
658 views

Continuum Hypothesis in formalized language. [closed]

The Continuum Hypothesis was advanced by Georg Cantor in 1878, before that Zermelo–Fraenkel set theory was stablished. "There is no set whose cardinality is strictly between that of the integers and ...
0
votes
0answers
42 views

Creating mathematics vs building houses

I found the following quote in the book "Calculus" by Michael Spivak. (At the first page of Part 5,Epilogue, where he will discuss fields, construction of the real number, and uniqueness of the real ...
1
vote
1answer
75 views

Proof that the minimal inductive set does not contain any more elements than $\{\emptyset, s(\emptyset), s(s(\emptyset)),s(s(s(\emptyset))),…\}$?

I am studying axiomatic (ZF) set theory and in particular the construction of the natural numbers from axiomatic set theory. Let me start by giving some introduction to my question. The axiom of ...
1
vote
0answers
38 views

Prob. 7, Chap. 1 in Baby Rudin

Here's problem 7 in the exercises following Chap. 1 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Fix $b > 1$, $y > 0$, and prove that there is a unique real number ...
1
vote
2answers
43 views

Can the Recursion Theorem be proved in Peano Arithmetic?

A recursive function on $\mathbb{N}$ can be defined as follows: Given an element $a \in \mathbb{N}$ and a function $f:\mathbb{N}\rightarrow\mathbb{N}$, we can define a function ...
6
votes
1answer
80 views

Prob. 6, Chapter 1 in Baby Rudin [duplicate]

Here's problem $6$ in Chapter $1$ in the book Principles of Mathematical Analysis by Walter Rudin, $3$rd edition: Fix a real number $b$, such that $b > 1$. $(a)$ If $m, n, p, q$ are integers, ...
0
votes
1answer
21 views

Step 6, Appendix to Chapter 1 in Baby Rudin: How to show that if $\alpha > 0^*$ and $\beta > 0^*$, then $\alpha \beta > 0^*$?

I'm reading Appendix to Chapter 1 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition. In this appendix, Rudin gives a proof of Theorem 1.19 by constructing $\mathbb{R}$ from ...
6
votes
1answer
81 views

How can one prove the axiom of collection in ZFC without using the axiom of foundation?

Say I want to prove the axiom(s) of collection from the axiom(s) of replacement. If you have the axiom of foundation, then you can use Scott's trick to do this. But suppose I'm working in a context ...
4
votes
2answers
63 views

Paradoxes that remain paradoxical even when you understand the underlying theory

It strikes me that the Banach-Tarski paradox (rearranging ball partitions) is not dispelled even when you understand the underlying mathematics. Perhaps Parrondo's paradox in Game Theory (sawtooth ...
0
votes
0answers
66 views

Algorithms in formal logic + ZFC

From my understanding, most of mathematics can be built up assuming a mechanical procedure of manipulating finite strings of symbols according to certain rules. A conventional way to do it is via the ...
2
votes
0answers
40 views

Is it possible to work entirely with Axioms and Isomorphisms?

I've been thinking a lot about foundations, again, and specifically how to make isomorphism invariance clearer - which seems to be one of the main purposes of alternative foundations such as Homotopy ...
3
votes
2answers
88 views

Definition of “set” in HoTT

In the homotopy type theory book (https://hott.github.io/book/nightly/hott-online-1007-ga1d0d9d.pdf) "set" is defined as follows (see chapter 3): Definition 3.1.1. A type A is a set if for all x, y : ...
3
votes
1answer
95 views

Where do models of set theory live?

When we are studying independence proofs we are dealing with statements of the form $Cons(T)\rightarrow Cons(T')$ where $T$ and $T'$ are first order theories; commonly $T$ and $T'$ are subtheories of ...
0
votes
0answers
43 views

Predicative definition of the natural number set in a Complete Field

Usually inside a Complete Field CF, the natural number set in a Complete Field $(\Bbb N_{CF})$ is defined as intersection of all inductive sets in CF, where the inductive set definition is as fallow: ...
0
votes
1answer
50 views

Some questions regarding some statements of Shoenfield.

Consider the system of arithmetic $\mathrm N$ found in Shoenfield's textbook Mathematical Logic: N1. Sx$\neq$0, N2. Sx=Sy$\rightarrow$x=y N3. x+0=x N4. x+Sy=S(x+y) N5. x$\cdot$0=0 N6. ...
1
vote
0answers
17 views

Is Grzegorczyk's theory $TC$ interpretable in Robinson Arithmetic $Q$

The question is in the title. It is known that $TC$ interprets Robinson Arithmetic $Q$ (Svejdar proved this), but I am interested in seeing the proof of the other direction. My motivation for the ...
4
votes
1answer
115 views

Model of homotopy type theory in ZFC

There is a model of ZFC in homotopy type theory Does exist a model of homotopy type theory in ZFC? Is there a proof of "equal logical expressivity" of these theories? p.s. I use word "model" in ...
1
vote
4answers
87 views

Reference request for set-theoretic foundations of geometry

My question is, Is it possible to define geometrical concepts (say, of Euclidean Geometry) like 'point', 'striaght line' in purely set theoretic terms? So far, I could think of the following ...
5
votes
2answers
401 views

Programming and ZFC

Suppose I have a simple program that implements an algorithm (say depth-first search), written in a simple imperative programming language with the standard for loops, recursions, conditional ...
7
votes
1answer
281 views

Are the addition and multiplication of real numbers, as we know them, unique?

After recently concluding my Real Analysis course in this semester I got the following question bugging me: Is the canonical operation of addition on real numbers unique? Otherwise: Can we define ...
2
votes
1answer
87 views

Where am I wrong in this “proof” that the collection of sets is countable?

Looking at sets (in ZF, Set Theory of Kunen) I could not escape from looking at logic as well. A formal language is presented containing basic symbols ($\wedge,\neg,\exists,(,),\in,=,$ and $v_{i}$ ...
3
votes
3answers
65 views

Can sets have derivatives?

I know how functions can be described, e.g. $y=x^2$ and in high school they teach you the general form, e.g. $x^2-y=0$ I believe so that later on they can abstract the notion of the curve into a ...
12
votes
6answers
838 views

Quantifier: “For all sets”

I've seen the following statement a few times: "Let $A$ be a set, then $\emptyset\subseteq A$". Or, written 'more formally': $$ \forall A\,\, \emptyset\subseteq A $$ My doubt is: I've ...
0
votes
0answers
23 views

Generalized Geometry and Analysis

Algebra and logic can be formalized in the very general frameworks of category theory and categorical logic. Can the same be done for geometry or analysis?
0
votes
0answers
28 views

References for the construction of various number types

Hello I am a high school student currently reading through Calculus by Spivak which has been recommended by many people on this site. I was slightly disappointed by the first chapter in which ...
1
vote
2answers
68 views

Extracting an infinite subsequence

Suppose that $\{a_i\}_{i\in\Bbb N}$ is a sequence of real numbers such that for any $i\in\Bbb N$, there exists $j\in\Bbb N$ with $j>i$ and $a_j>a_i$. How to prove that $\{a_i\}$ contains an ...
0
votes
1answer
32 views

Functional definition of tuples

As I was reading through some wiki articles, as well as my texts, I found the following two definitions for the ordered pairs and $n$-tuples: $(a,b):=\{\{a\},\{a,b\}\};\quad ...
-2
votes
1answer
60 views

What is the importance of the axiomatization of set theory? [closed]

Well I know that de axiomatization is important to establish certain laws, etc. But what other arguments do exist?
39
votes
4answers
2k views

Why are we justified in using the real numbers to do geometry?

Context: I'm taking a course in geometry (we see affine, projective, inversive, etc, geometries) in which our basic structure is a vector space, usually $\mathbb{R}^2$. It is very convenient, and also ...
1
vote
0answers
24 views

To what extent is the presentation of the category of X a presentation of the theory of X?

Various categories have been axiomatized. To what extend are these axiomatizations acceptable as presentations of the theories of those categories? For instance, is Lawvere's "An Elementary Theory ...
6
votes
6answers
403 views

When using an axiom scheme, are we implicitly using a choice principle?

I heard an interesting argument from a colleague recently that went something like this. Whenever we are using an axiom scheme, we are essentially choosing one of the instances of this scheme, and ...
3
votes
0answers
56 views

Category theory without sets

I have been reading Mac Lane's Categories for the Working Mathematician, and the prospect of developing category theory without any use of set theory is mentioned more than once in the book, but never ...
4
votes
1answer
69 views

Proving that the existence of strongly inaccessible cardinals is independent from ZFC?

ZFC can't prove that strongly inaccessible cardinals exist, or else it would prove that a model of itself exists and hence $Con(ZFC)$. So this leaves us with two options: ZFC proves there are no ...
1
vote
1answer
65 views

Intensional vs. extensional equality (or something like this)

I'm reading this thesis by Michael Warren on Homotopy Type Theory. I'm really a newbie to the field and got puzzled right in the beginning, where the following rule appears: A little bit before he ...
0
votes
0answers
47 views

Was Ramsey mistaken in thinking that the same proposition can be both elementary and non-elementary in form?

According to Ramsey's Foundations of Mathematics, chapter III, suppose $'a', 'b', ..., 'z'$ were all the individuals, then $\phi{a}.\phi(b)...\phi(z)$ expresses the same proposition as $(x)\phi(x)$ ...
41
votes
6answers
2k views

Is it possible to formulate category theory without set theory?

I have never understood why set theory has so many detractors, or what is gained by avoiding its use. It is well known that the naive concept of a set as a collection of objects leads to logical ...
2
votes
2answers
80 views

What is the right model of this set?

I am doing a research in foundation of maths, especially in logic area. Right now, I am building a set $\mathbb{L}$ through this five axioms: Additive axiom: closeness, commutative, associative, 0 ...
1
vote
1answer
25 views

The diagonal product of the family $\{f_\alpha:\alpha\in I\}$ is injective.

Suppose that $f_\alpha : X\longrightarrow Y_\alpha$ for any $\alpha\in I$, where $X$,$Y_\alpha$ be nonempty sets for all $\alpha\in I$, and Let $f : X\longrightarrow \prod\{Y_\alpha: \alpha\in I \}$ ...
3
votes
2answers
123 views

Minimal foundations for Cardinal Arithmetic

I would like to develop a theory of cardinal numbers that relies on as weak a basis as possible. Therefore, I would like to know if there is a way to even define a cardinal number for every set ...
2
votes
1answer
55 views

$\mathbb{Z}$: Unique operation s.t. $1\star m=m$ and right distributivity hold?

Given $\mathbb{Z}$ and the usual addition $+$ on it, do we have unicity of a binary operation $\star$ such that \begin{align*} \tag{1}1\star m&=m\\ \tag{2}(m+n)\star p&=m\star p+n\star p ...
0
votes
1answer
17 views

Multiplication in $\mathbb{N}$ from two simple laws

From Barry Mazur's Imagining Numbers: (particularly the square root of minus fifteen), p. 97: Suppose that you have some unknown operation that allows, as its "input," any two positive whole ...
2
votes
0answers
51 views

Is there any solution to Frege's criticisms of Hilbert's Geometry without the application of Model Theory?

Recently I have come across the interesting debate of Frege and Hilbert regarding the Foundations of Geometry. It seems to me that the main concern of Frege was on the Logical Consistency of Hilbert's ...
2
votes
1answer
82 views

Is math fool-proof? [duplicate]

I am an high school student, and I'm diving deeper and deeper into Maths, and thinking into studying it at university. I have read multiple books, and gazed at the beautiful proofs presented there. In ...
2
votes
2answers
92 views

How to prove induction principle by real number axioms?

In axiomatic approach to real numbers, that is by defining them to be the complete ordered field, one is expected to prove every theorem and solve every problem by using ultimately only the axioms. I ...
0
votes
0answers
25 views

Reworked Induction Problem

The is a reworked problem from a previous post I corrected. Think I have it, but any pointers, corrections, criticisms are welcome. Thanks in advance. 1³+2³+···+n³ < ½n⁴ , ∀n∈ℕ, n≥3 For n=1, ...
46
votes
13answers
4k views

What is the definition of a set?

From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not $0^0$ is $1$ is a simple matter of definition. My ...